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As an alternative, you can specify base DNS servers for resolvconf at /etc/resolvconf/resolv.conf.d/base, which won't be overwritten. It uses the same syntax. That way you can keep running Ubuntu or Mint the default way. Then tell resolvconf to regenerate resolv.conf with sudo resolvconf -u.

Hover, you can remove resolvconf:
sudo apt-get remove resolvconf. It is safe to do so. Then write your desired DNS resolvers into /etc/resolv.conf.

You may not be able to use DNS management features in other programs that automatically manage your network (networkmanager) anymore.

If you're uncomfortable with removing the resolvconf package, remove /etc/resolv.conf (on systems with the resolvconf package it will be a symlink to /run/resolvconf/resolv.conf) and create the file with the contents you desire.
– DeathgripJun 28 '17 at 16:27

@Deathgrip: That's what I've done before, I but couldn't remember what it was that I did. Removing resolvconf is also a good solution.
– NickJul 3 '17 at 0:09